2007/01/01 by Dragos Iftimie, Dragoş Iftimie, Genevieve Raugel +2 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · doi:10.1512/iumj.2007.56.2834
crossref issued 2007/01/01 · crossref published 2007/01/01 · crossref published-print 2007/01/01 · openalex publication_date 2007/01/01 · crossref created 2007/04/26 · crossref deposited 2011/06/15 · openalex created_date 2025/10/10 · crossref indexed 2026/07/28 · openalex updated_date 2026/07/29
Abstract. We consider the Navier-Stokes equations on a thin domain of the form Ωε = x ∈ R3; x1, x2 ∈ (0, 1), 0 < x3 < εg(x1, x2) supplemented with the following mixed boundary conditions: periodic boundary conditions on the lateral boundary and Navier boundary conditions on the top and the bottom. Under the assumption that ‖u0‖H1(Ωε) ≤ Cε− 1 2, ‖Mui0‖L2(Ωε) ≤ C for i ∈ 1, 2 and similar assumptions on the forcing term, we show global existence of strong solutions; here ui0 denotes the i-th com-ponent of the initial data u0 and M is the average in the vertical direction, that is, Mui0(x1, x2) = 1 ε g ∫ ε g 0 ui0(x1, x2, x3) dx3. Moreover, if the initial data, respectively the forcing term, converge to a bidimensional vector field, respectively forcing term, as ε → 0, we prove convergence to a solution of a limiting system which is a Navier-Stokes-like equa-tion where the function g plays an important role. Finally, we compare the attractor of the Navier-Stokes equations with the one of the limiting equation.